Let \(f\in C(\mathcal {A})\)  with \(\Vert f\Vert \leqslant 1\) . We define \(\text{Act}(f,b,n):=M^{n}f(b)=\sum _{c\in \varrho ^{n}(b)} f(c)\)  and \(\text{Exp}(f,b,n)=\lambda ^n\cdot \ell (b) \operatorname{dens}()\) .
(Later we will consider \(f(b)=1\)  for all \(b \in \mathcal {A}\) . Then
\(\text{Act}(f,b,n)\)  is the actual number of tiles in \(\varrho ^n_{a}(b)\) , and \(\text{Exp}(f,b,n)\)  is the expected number of tiles in \(\varrho ^n_{a}(b)\) .)
The following discrepancy estimate follows from [1]}.
